During a research experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 7:00am there were 6,000 bacteria present in the culture. At noon, the number of bacteria grew to 6,700. How many bacteria will there be at midnight?
If the growth rate is proportional to the population, then it is exponential. That is the number of bacteria at time t, call it B(t)=B(0)ekt. In our case we have B(t)=6000ekt. The k is determined by noting that 12 noon, t=5 we have 6700=6000e5k, and so 5k=ln(6700/6000), or k=.022069611433773. At midnight, t=17 hours we have that
B(17)=6000e.37518339...=6000×1.45525827...=8731.549... or about 8732